4.1 Introduction

This chapter presents the proposed for transmitting images over a noisy channel. The proposed method employs OFDM and chaotic encoder schemes to achieve distortion less and secure image transmission over the noisy channel. The latter part of this chapter discusses the brief overview of the tool used i.e., Mat-lab image processing and wavelet toolbox.

4.2 Problem Identification

The main do research has resulting from the following problems has been identified:

• There is need to address the tradeoff between the power consumption and the quality of service in wireless media systems.

• Digital images are attractive data types with widespread range of use and many user are stimulating to make it secure.

• Transmission of images over varying channel conditions.

• Single OFDM transmitter.

• Limited power resources.

4.3 Problem Identification Solution

The Proposed system employs the combination of OFDM (Orthogonal Frequency Division Multiplexing) and Chaotic Communication.

4.3.1 What is orthogonal frequency-division multiplexing?

Orthogonal frequency-division multiplexing (OFDM) is a method of encoding digital data on multiple carrier frequencies. OFDM has developed into a popular scheme for wideband digital-Communication, used in applications such as digital television and audio broadcasting, DSL Internet access, wireless networks, power line networks, and 4G mobile communications

OFDM is a frequency-division multiplexing (FDM) scheme used as a digital multi-carrier modulation method. A large number of closely spaced orthogonal sub-carrier signals are used to carry data on several parallel data streams or channels. Each sub-carrier is modulated with a conventional modulation scheme (such as quadrature amplitude modulation or phase-shift keying) at a low symbol rate, maintaining total data rates similar to conventional single-carrier modulation schemes in the same bandwidth.

The primary advantage of OFDM over single-carrier schemes is its ability to cope with severe channel conditions.

For example, attenuation of high frequencies in a long copper wire, narrowband interference and frequency-selective fading due to multipath without complex equalization filters.

Channel equalization is simplified because OFDM may be viewed as using many slowly modulated narrowband signals rather than one rapidly modulated wideband signal. The low symbol rate makes the use of a guard interval between symbols affordable, making it possible to eliminate inter symbol interference (ISI) and utilize echoes and time-spreading (on analogue TV these are visible as ghosting and blurring, respectively) to achieve a diversity gain, i.e. a signal-to-noise ratio improvement. This mechanism also facilitates the design of single frequency networks (SFNs), where several adjacent transmitters send the same signal at the same moments in time at the same frequency, as the signals from multiple distant transmitters may be combined constructively, rather than interfering as would typically occur in a traditional single-carrier system.

OFDM Advantages

• Makes efficient use of the spectrum by allowing overlap.

• By dividing the channel into narrowband flat fading sub channels, OFDM is more resistant to frequency selective fading than single carrier systems are.

• Eliminates ISI and IFI through use of a cyclic prefix.

• Using adequate channel coding and interleaving one can recover symbols lost due to the frequency selectivity of the channel.

• Channel equalization becomes simpler than by using adaptive equalization techniques with single carrier systems.

• It is possible to use maximum likelihood decoding with reasonable complexity.

• OFDM is computationally efficient by using FFT techniques to implement the modulation and demodulation functions.

• Is less sensitive to sample timing offsets than single carrier systems are.

• Provides good protection against co channel interference and impulsive parasitic noise.

OFDM disadvantages

• The OFDM signal has a noise like amplitude with a very large dynamic range; therefore it requires RF power amplifiers with a high peak to average power ratio.

• It is more sensitive to carrier frequency offset and drift than single carrier systems are due to leakage of the DFT.

4.3.2 Why using chaotic communication?

Chaotic communications is an application of chaos theory which is aimed to provide security in the transmission of information performed through telecommunications technologies. By secure communications, one has to understand that the contents of the message transmitted are inaccessible to possible eavesdroppers.

In chaotic communications security (i.e., privacy) is based on the complex dynamic behaviors provided by chaotic systems. Some properties of chaotic dynamics, such as complex behavior, noise-like dynamics (pseudorandom noise) and spread spectrum, are used to encode data. On the other hand, chaos being a deterministic phenomenon, it is possible to decode data using this determinism. In practice, implementations of chaotic communications devices resort to one of two chaotic phenomena: synchronization of chaos, or control of chaos.

To implement chaotic communications using such properties of chaos, two chaotic oscillators are required as a transmitter (or master) and receiver (or slave). At the transmitter, a message is added onto a chaotic signal and then, the message is masked in the chaotic signal. As it carries the information, the chaotic signal is also called chaotic carrier. Synchronizing of these oscillators is similar to synchronizing random neural nets in neural cryptography.

When chaos synchronization is used, a basic scheme of a communications device (Cuomo and Oppenheim 1993) is made by two identical chaotic oscillators. One of them is used as the transmitter, and the other as the receiver. They are connected in a configuration where the transmitter drives the receiver in such a way that identical synchronization of chaos between the two oscillators is achieved. For the purpose of transmission of information, at the transmitter, a message is added as a small perturbation to the chaotic signal that drives the receiver. In this way, the message transmitted is masked by the chaotic signal. When the receiver synchronizes to the transmitter, the message is decoded by a subtraction between the signal sent by transmitter and its copy generated at the receiver by means of the synchronization of chaos mechanism. This works because, whilst the transmitter output contains the chaotic carrier plus the message, the receiver output is made only by a copy of the chaotic carrier without the message.

It is due to the dynamic, non-linear and unpredictable properties the real world exhibits. It is difficult to model it using a simple mathematical formula. Only part of the real world myth can be solved with these types of systems, thus requires a more effective and accurate solution. A solution that reflects the unpredictable nature of our world is the “Chaos Theory”. It provides the required kind of system behavior (non-linear, dynamic, unpredictable and etc), thus it has been widely studied by mathematicians and scientists alike.

4.4 Proposed work

Fig. 4(a) below shows block diagram of the transmission section of the proposed system. The input image is first converted into bits and then these bits are given to different chaotic encoder blocks. The output of the encoders is then given to the OFDM respective transmitters section. The OFDM outputs are then transmitted using the different antennas of the MIMO system as shown in Fig. 4(b).

Fig. 4(a): Transmitter Section of MIMO system

Fig. 4(b): Receiver Section of MIMO system

At the receiver side, the receiving antennas receiver the transmitted image copies. The received bits are first given to the OFDM receiver, the output of which is given to the chaotic decoder. The mean of the outputs from the chaotic decoders is calculated the finally converted to give the final output image as in Fig. 4(b). Since the proposed technique uses employs a MIMO system the transmitted copies of images at different input power levels undergo different distortion through the channel. At the receiver these image copies are decoded and a mean is calculated to give the output image. This minimizes the distortion in the received image.

4.5 Proposed methodology

Unequal power allocation can be performed in real-time using well developed optimization techniques. But solution might not be global. Due to the complex nature of the expressions for MSE, it is mathematically intractable to derive a closed form solution to the power optimization problem. There are many well developed techniques to obtain numerical solutions to such optimization problems. Here, the Kuhn-Tucker equations along with a sequential quadratic programming (SQP) method are used to solve this constrained multivariable minimization problem. The SQP method formulates and solves a quadratic programming (QP) sub-problem at all iteration of the optimization process.

This method employs the Broyden-Fletcher-Goldfarb-Shannon (BFGS) formula to estimate the Hessian of the Lagrangian at all iteration. An active set strategy similar to that described in [36] is used to solve the QP sub problem. To solve this SQP problem, MATLAB’s optimization toolbox is used. Using this method, the optimum MSE and the corresponding transmission power vector are obtained. An interesting thing to note is that at any given instant, the channel from a particular transmit antenna to the receive antennas might be better than the channel corresponding to the remaining transmit antennas. In fact, the channels from different transmit antennas to the receive antennas are very likely to be different at different times.

Therefore, a natural idea is to transmit more important streams from “more reliable” transmit antennas and less important streams from “less reliable” antennas. This makes sense intuitively since less power will be required by the most important stream if it is being transmitted from the best antenna as compared to that of a random antenna. Hence, more power can be allocated to less important streams resulting in further reduction of overall distortion. Since the channel stays constant for a block of symbols, and then changes, an antenna selection process needs to be performed for each block of symbols in real time. Antenna selection is a research problem of its own and there is a large amount of literature available on this topic. Instead of using any sophisticated antenna selection methods that are available, a very simple method of antenna selection based on SINR is used to keep the optimization problem simple and computationally less intensive.

At any channel instantiation, first the four SINRs for the four streams are computed for the case of equal power allocation. Then, the transmit antenna corresponding to the stream with highest SINR is selected to transmit the most important stream, the transmit antenna with the second highest SINR to transmit the second most import stream and so on. This method of antenna selection is static as it assigns different antennas to different streams at the beginning of the optimization procedure for each channel instantiation based on the equal power case. Though this schemes will give us the second best antenna and so on.

This is because the SINR for the streams transmitted from different antennas changes when the transmit power is varied between antennas, which in turn can change the order of the best to worse SINR streams, hence making another antenna the second best rather than the one found initially, in terms of SINR. A better scheme would be to assign transmit antennas dynamically during the optimization procedure, however, that will increase the computational complexity since more iterations would be needed. Nevertheless, as observed by simulations, this antenna selection schemes does give significantly better results than that of randomly assigning antennas to different streams.

Antenna selection does not create any problem at the receiver since the receiver computes the received SINR for each stream and hence discovers the order of importance of the streams. After antenna selection, constrained power optimization is performed iteratively by searching through different combinations of transmission power allocation to different streams. MSE is computed for these different combinations of transmission power and the power allocation vector corresponding to minimum the MSE is chosen. The total transmit power at any given instant is kept constant. Note that the main goal of problem is to demonstrate that significant quality gains can be achieved by using unequal power allocation matched to image statistics in a MIMO system. Once the problem is formulated, well established optimization algorithms can be used to find the optimal solution. As discussed above, a SQP method is used to find the minimum MSE and the corresponding transmission power allocation scheme. However, like most of the numerical optimization methods, this method is also computationally extensive. To reduce the number of computations performed, a very simple suboptimal power allocation method is proposed in the following section.

We can calculate MSE and SINR by using following equations,

……………………(1)

In equation no. (1) Where is the column and row entry of . This SINR can be easily related to the instantaneous BER for 4-QAM using the following expression:

………………….(2)

Where Q is the Q function. Using these relations between SINR, BER, . The MSE can be related to the transmission power (energy) .

………………….(3)

Where is a set containing pairs of segment number and partition number contained in stream of block number n. Note that the portioning of segments into different blocks depends on the total number of bits in the image, and the block length.

4.6 Antenna selection method

A natural idea is to transmit more important streams from “more reliable” transmit antennas and less important streams from “less reliable” antenna. This makes sense intuitively since less power will be required by the most important stream if it is being transmitted from best antenna as compared to that of random antenna. Hence, more power can be allocated to less important streams resulting in further reduction of overall distortion.

Antenna selection is a research problem of its own instead of using any of the sophisticated antenna selection methods that are available, a very simple method of antenna selection based on SINR (Signal to Interference and Noise Ratio) is used to keep the optimization problem simple and computationally less intensive.

4.6.1 Flow chart on MIMO system

Fig. 4(c): Flow chart on MIMO System

There are send to uncompress image in sender. And there are send to image in jpeg encoder. And apply antenna selection method to send sender. Then antenna select image convert to UPA form. By using suboptimal algorithm apply UPA then send to chaotic encoder. It’s secure and reliable in chaos theory. And then send encoded image OFDM, transmitter by using multiple inputs and multiple output. Then received OFDM received by multiple input and multiple output. then received the chaotic decoder. And by using suboptimal algorithm apply UPA. And received the apply antenna selection method. Then received the jpeg decoder and then used in form of uncompress image.

4.7 Suboptimal problem algorithm

Our original optimization problem is a minimization problem in four variables. Most numerical optimization methods are computationally intensive for optimization problems with more than two variables. For real-time applications, it is necessary that the power optimization procedure should be computationally non-intensive. The number of computations can be significantly reduced by devising simple suboptimal algorithms that divide the original problem into optimization problems with fewer numbers of variables, without imposing a large penalty on performance. Based on this idea, a suboptimal algorithm for the power allocation problem is developed. This algorithm quantizes the transmit power for different streams and essentially breaks down the four variable optimization problem into an iterative two variable optimization problem.

After performing antenna selection the range of transmit power for each stream is quantized in levels, where k=1 corresponds to the most important and k=4to the least important stream. The algorithm starts by setting the initial minimum MSE ( to a very large value (infinity), the total available power to and total allocated power to to zero. The algorithm then varies the transmit power for the 1st stream in step of from to , while varying the transmit power(energy) for stream 2 to 4 equally in step of . The main idea here is to vary the power in steps while dividing the remaining power equally between the remaining three layers.

The algorithm computes MSE at each step, and if the MSE is lower than the previous MSE at each step, and if the MSE is lower than previous , it updates to this value. The computations for the 1st stream are stopped when either the entire range of available power has been spanned or when the SINR for any of the three streams. The minimum MSE of all these combinations is then assigned to and the corresponding transmission power for stream 1 is fixed . The allocated power is modified to and the same process is repeated for the remaining streams, the transmission power for the 1st to streams are fixed (already found), the transmit powers for the stream is varied in steps of , and the transmit power for streams are varied equally in steps of . This way, at any given time the optimization problem is essentially a two variable constrained minimization problem, hence reducing the computational complexity significantly.

This algorithm is summarized below:

Initialize: k=1, m=1, , , = /

Step 1: Do

,

Find MSE(

If MSE( <

Then MSE( ,

m=m+1.

While m< AND

Step 2: ,

If k<2 then go to step 1,

Else has the minimum value of MSE, and has the corresponding transmit power for different streams.

This algorithm uses the fact that received SINR for more important stream needs to be greater than the received SINR for a less important stream to minimize the distortion.

4.8 Overview of Mat-lab

MATLAB is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. Typical uses include:

• Math and computation

• Algorithm development

• Modeling, simulation, and prototyping

• Data analysis, exploration, and visualization

• Scientific and engineering graphics

• Application development, including graphical user interface building

MATLAB has evolved over a period of years with input from many users. In university environments, it is the standard instructional tool for introductory and advanced courses in mathematics, engineering, and science. In industry, MATLAB is the tool of choice for high-productivity research, development, and analysis. The key feature of MATLAB includes:

• High-level language for technical computing

• Development environment for managing code, files, and data

• Interactive tools for iterative exploration, design, and problem solving

• Mathematical functions for linear algebra, statistics, Fourier analysis, filtering, optimization, and numerical integration

• 2-D and 3-D graphics functions for visualizing data

• Tools for building custom graphical user interfaces

• Functions for integrating MATLAB based algorithms with external applications and languages, such as C, C++, FORTRAN, Java™, COM, and Microsoft® Excel.

Toolboxes MATLAB features a family of application-specific solutions called toolboxes. Very important to most users of MATLAB, toolboxes allow you to learn and apply specialized technology. Toolboxes are comprehensive collections of MATLAB functions (M-files) that extend the MATLAB environment to solve particular classes of problems. Areas in which toolboxes are available include signal processing, control systems, neural networks, fuzzy logic, wavelets, simulation, and many others.

The MATLAB system consists of five main parts:

• Development Environment. This is the set of tools and facilities that help you use MATLAB functions and files. Many of these tools are graphical user interfaces. It includes the MATLAB desktop and Command Window, a command history, and browsers for viewing help, the workspace, files, and the search path.

• The MATLAB Mathematical Function Library. This is a vast collection of computational algorithms ranging from elementary functions like sum, sine, cosine, and complex arithmetic, to more sophisticated functions like matrix inverse, matrix Eigen values, Bessel functions, and fast Fourier transforms.

• The MATLAB language. This is a high-level matrix/array language with control flow statements, functions, data structures, input/output, and object-oriented programming features. It allows both “programming in the small” to rapidly create quick and dirty throw-away programs, and “programming in the large” to create complete large and complex application programs.

• Handle Graphics- This is the MATLAB graphics system. It includes high-level commands for two-dimensional and three-dimensional data visualization, image processing, animation, and presentation graphics. It also includes low-level commands that allow you to fully customize the appearance of graphics as well as to build complete graphical user interfaces on your MATLAB applications.

• The MATLAB Application Program Interface (API). This is a library that allows you to write C and FORTRAN programs that interact with MATLAB. It include facilities for calling routines from MATLAB (dynamic linking), calling MATLAB as a computational engine, and for reading and writing MAT-files.

4.8.1 Image processing toolbox

Image Processing Toolbox provides a comprehensive set of reference-standard algorithms and graphical tools for image processing, analysis, visualization, and algorithm development. You can perform image enhancement, image deploring, feature detection, noise reduction, image segmentation, geometric transformations, and image registration. Many toolbox functions are multithreaded to take advantage of multicourse and multiprocessor computers. Image Processing Toolbox supports a diverse set of image types, including high dynamic range, gig pixel resolution, embedded ICC profile, and topographic. Graphical tools let you explore an image, examine a region of pixels, adjust the contrast, create contours or histograms, and manipulate regions of interest (ROIs). With toolbox algorithms you can restore degraded images, detect and measure features, analyze shapes and textures, and adjust color balance.

• Key Features

Image enhancement, filtering, and deploring

Image analysis, including segmentation, morphology, feature extraction, and measurement

Spatial transformations and image registration

Image transforms, including FFT, DCT, Radon, and fan-beam projection

Workflows for processing, displaying, and navigating arbitrarily large images

Modular interactive tools, including ROI selections, histograms, and distance measurements

ICC color management

Multidimensional image processing

Image-sequence and video display

DICOM import and export

4.8.2 Wavelet toolbox

Wavelet Toolbox software extends the MATLAB technical computing environment with graphical tools and command-line functions for developing wavelet-based algorithms for the analysis, synthesis, de-noising, and compression of signals and images. Wavelet analysis provides more precise information about signal data than other signal analysis techniques, such as Fourier.

The Wavelet Toolbox supports the interactive exploration of wavelet properties and applications. It is useful for speech and audio processing, image and video processing, biomedical imaging, and 1-D and 2-D applications in communications and geophysics.

• Key Features

Standard wavelet families, including Daubechies wavelet filters, complex Morley and Gaussian, real reverse bi-orthogonal, and discrete Meyer

Wavelet and signal processing utilities, including a function to convert scale to frequency

Methods for adding wavelet families

Lifting methods for constructing wavelets

Customizable presentation and visualization of data

Interactive tools for continuous and discrete wavelet analysis

Wavelet packets, implemented as MATLAB objects

One-dimensional multi signal analysis, compression, and de-noising

Multi scale principal component analysis

Multivariate demising

Analyzing Signals and Images the Wavelet Toolbox graphical user interface (GUI) provides a comprehensive set of tools for analyzing 1-D and 2-D signals, including tools for wavelet analysis, wavelet packet analysis, de-noising, and compression.

For 1-D signals, you can use the GUI tools to:

• Perform discrete wavelet analysis of signals.

• Perform continuous wavelet analysis of real signals using complex wavelets.

• De-noise signals.

• Estimate wavelet-based density.

• Perform wavelet reconstruction schemes based on various wavelet coefficient selection strategies.

• Randomly generate fractional Brownian motion.

• Perform 1-D signal extension and truncation using periodic, symmetric, smooth, and zero padding methods.

• Perform 1-D signal clustering and classification using wavelet analyses (with Statistics Toolbox, available separately).

For 2-D signals, you can use the GUI tools to:

• Perform discrete wavelet analysis of images

• Fuse two images

• Perform translation-invariant de-noising of images, using the stationary wavelet transform. Reconstruct wavelet schemes based on various wavelet coefficient selection strategies.

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